Peculiar properties of hermitian and positive n-times integrated C-cosine functions on Banach spaces are investigated. Among them are: (1) Any nondegenerate positive n-times integrated C-cosine function is infinitely differentiable in operator norm; (2) An exponentially bounded, nondegenerate C-cosine function on L-P(mu)(1 < p < infinity) (or L-1(mu), C-0, in case C has dense range) is positive if and only if its generator is bounded, positive, and commutes with C.
